Mathematician Levent Alpöge made a surprising announcement on X, revealing that he had used Anthropic's Fable 5 to disprove the Jacobian conjecture, a long-standing open problem in algebraic geometry. This achievement has sparked debate and discussion within the mathematical community, with experts weighing in on its implications. While some view it as a significant milestone in AI's ability to tackle complex mathematical problems, others argue that it's more about the power of computational search than a true breakthrough. The Jacobian conjecture, included in Smale's problems, has eluded mathematicians for nearly 90 years, and Alpöge's use of Fable 5 to find a counterexample is a notable feat. However, the impact of this discovery is still a matter of debate. Professor Andrew Blumberg, an expert in mathematics and AI, suggests that the counterexample itself doesn't provide much insight, as it's a result of computational search rather than a deep understanding of the problem. He compares it to finding an answer without learning from it, emphasizing the importance of positive proofs that lead to new knowledge. The recent success of AI in disproving the Jacobian conjecture follows a similar pattern to its achievement in disproving the Erdős unit distance conjecture. However, Blumberg notes that the latter was more productive, as it led to further exploration and understanding of the problem. The use of AI in mathematics raises questions about the nature of mathematical discovery and the role of computational tools in advancing our understanding of complex problems. As AI continues to evolve, it will be fascinating to see how it shapes the future of mathematical research and whether it can truly replicate the creative and insightful aspects of human mathematical thinking.
